Ophthalmic lens for presbyopia and aphakia

ABSTRACT

There is presented a multifocal ophthalmic lens of homogeneous transparent optical material useful for the correction of the refractive error and the accommodative insufficiency or absence of accommodation in presbyopia and in aphakia, the lens characterized by having a unique variable front surface, and a coacting conicoid surface of eccentricity zero or greater, or toric, back surface, said variable front surface characterized by being geometrically and optically regular and continuous and having a pair of intersecting orthogonal principal planes, the first of said planes, generally horizontal, intersecting said variable surface normally at all points in a circular or elliptical or substantially elliptical great arc, the derivative of curvature vanishing at said great arc in sections orthogonal to it; the second of said planes, generally vertical, intersecting said variable surface normally at all points in a principal curve, about which there is symmetry, at least the portion of said principal curve below said great arc increasing in curvature progressively, said orthogonal principal planes intersecting along the axis of said variable surface, said axis intersecting said variable surface normally at an axial umbilical point where the derivative of curvature vanishes, all transverse sections of said surface at least below said great arc by planes orthogonal to said principal curve being conics of eccentricity greater than zero, or slight modifications thereof, the axes of all of said conics intersecting normally said principal curve, and the apical curvatures of said conic transverse sections at said principal curve being substantially equal to and increasing at an accelerated rate substantially equal to the corresponding curvature and rates of change respectively along said principal curve, while the corresponding eccentricities of said conic transverse sections also increase.

This application is a continuation-in-part of my pending applicationSer. No. 322,488, filed Jan. 10, 1973, now abandoned.

This invention relates to an improved ophthalmic lens, either ofspectacle lens or contact lens type, primarily intended for the reliefof the refractive error and the accommodative insufficiency or absenceof accommodation in presbyopia and aphakia. Ordinarily the opticaltreatment of said insufficiency or absence of accommodation isaccomplished with either simple reading glasses, or bifocals, ortrifocals in spectacles, or with bifocals in corneal contact lenses. Inthe lens of this invention, the change in optical power of theophthalmic lens required to supplement the failing or absence ofaccommodation in presbyopia and the lack of accommodation in aphakia isaccomplished in a continuous and regular manner, without discontinuitiesin the field of vision through the lens, and with minimal distortion inthe field. The design of the lens in its principal embodiment is suchthat, in gaze at distant objects through the upper portion of the lensas it is worn in spectacles, or worn as a corneal contact lens, visionis clear, and as objects are observed through lower and lower portionsof the lens, they must be closer and closer to the wearer to be seenclearly. The continuous and regular increase in refractive power fromthe upper distance portion to the lower border results from thecombination of a conicoid back surface of eccentricity zero or greateror a toric back surface, the toric being used when ocular astigmatism isto be corrected, and a unique front surface which increases in curvaturecontinuously and regularly in an accelerated manner from said upperportion to provide the increasing refractive power for the correction ofthe accommodative insufficiency or absence of accommodation.

Throughout the remainder of this specification and in the clamis, Ishall speak of conics as sections of the unique front surface of thelens of this invention. I intend including within the definition of saidconics those slight modifications which are dependent upon and are aconsequence of the shape of the edge of a circular cam follower and theshape of the edge of a circular abrading tool which are used in themethod and apparatus of this invention to produce said unique frontsurface.

A conic can be described in terms of its focus, f, and eccentricity, e,that is, in terms of magnitude and shape. The eccentricity, e, of aconic is a constant and is given by the differential equation:

    e = df/dx                                                  (1)

where f is the focal radius of the conic and x is the coordinate alongthe axis containing said focus with the apex of the conic as the origin.Should df/dx vary with x, then a more appropriate mathematicaldescription of the resulting modified conic in terms of eccentricity cantake the form of a Taylor series which takes into account the rate ofchange of eccentricity. Using MacLaurin's formula: ##EQU1## where e_(g)given by equation (2) is defined as the generalized or effectiveeccentricity. When the derivatives of eccentricity are small, themodified conic can osculate a conic over a relatively large extent abouttheir common apex. Hence, for the purpose of simplifying the descriptionof this invention, both the osculating conic and the modified conic willbe described as of the focus and eccentricity, or of the apical radiusof curvature and eccentricity, of the osculating conic, it beingunderstood that differences in the two curves manifest themselves whenthe curves are extended.

For the purpose of simplifying the description of this invention, thepoints of intersection of the major and minor axes of an ellipse withthe ellipse will hereinafter be called the prolate and oblate pointsrespectively.

In the drawings:

FIG. 1 is a diagrammatic view of the lens of this invention showing thehorizontal and vertical principal planes cutting the variable surface ofthe lens in the great arc and principal curve respectively;

FIG. 2 is a diagrammatic and geometric view of the variable surface WQVPin the first embodiment of the lens of this invention showing theprincipal curve QBP and transverse sections of the variable surface ascircular arcs above the circular great arc WBV at R, S and T, and asconics at F, M and H below the great arc;

FIG. 3 is a diagrammatic view used to represent several embodiments ofthe variable surface WQVP of the lens of this invention. Arc QBP is theprincipal curve. Point B is the axial umbilical point. In oneembodiment, point B is the oblate point of elliptical arc BP and theprolate points of elliptical arc QB and elliptical great arc WBV. Inthis embodiment, all transverse sections of the variable surface areconics of eccentricity greater than zero with their apices along theprincipal curve, selected apical points shown at R, S, T, B (the axialumbilical point), F, M and H. Transverse sections through F, M and Hincrease successively in curvature at their apices and in eccentricity,the curvatures at the apices of said transverse sections or transversecurvatures being substantially equal to the corresponding verticalcurvatures at said points. In another embodiment, principal curve QBP isan elliptical arc with its oblate point at B, the axial umbilical point.Great arc WBV is circular. Transverse sections through F, M and H areconics which increase in curvature successively at apical points F, Mand H, and in eccentricity, the transverse curvatures through saidpoints being substantially equal to the corresponding verticalcurvatures. The remainder of the variable surface is identical to thatjust described, the surface being symmetrical about great arc WBV. Inanother embodiment, the two portions of the variable surface on oppositesides of the great arc WBV are similar to that just described but notidentical;

FIG. 4 is a diagrammatic view of one of the embodiments of the lens ofthis invention in which the variable surface WQVP has a circular greatarc WBV which is umbilical along its entire extent and along which thederivative of curvature of said surface vanishes. At each of points 1,2, 3, B, 4, 5 and 6 along said great arc, vertical and horizontalcurvatures are equal and curvatures at all of said points are equal inmagnitude;

FIG. 5 is a diagrammatic view of one of the embodiments of the lens ofthis invention in which the variable surface WQVP has an ellipticalgreat arc WBV with its prolate point at axial point B at which thevariable surface is umbilical and at which the derivative of curvaturevanishes. At each of points 1, 2, 3, B, 4, 5 and 6 along said great arc,the derivative of curvature of a vertical section orthogonal to saidgreat arc vanishes;

FIG. 6 shows an elliptical arc A'BA which provides a principal curve,major axis A'OA, semi-major axis OA, semi-minor axis OB and itsextension OG. Arc GC is the evolute or locus of centers of curvature forelliptical arc segment BA. GB is the radius of curvature of theelliptical arc at point B as well as the radius of curvature of thegreat arc of this invention at right angles to A'BA at point B. GB is,therefore, normal to both the principal curve and the great arc at B,the previously mentioned axial umbilical point;

FIG. 7 shows a plane section containing the axis OC of a right circularcone L'OL, of cone angle L'OL = 2ω, in the plane of the drawing, andsections through the cone perpendicular to the plane of the drawing at adistance l from the apex of the cone along element OL and various anglesφ with respect to that section perpendicular to said element OL of saidcone;

FIG. 8 shows a principal curve primarily intended for a cataract lensusing this invention formed of two elliptical portions, an upper portionQB with its prolate point at B, the axial point, and a lower portion BPwith its oblate point at B. Arc LGM is the combined evolute arc for saidtwo portions of the principal curve; LG the evolute arc related toelliptical arc QB, and GM the evolute arc related to elliptical arc BP;

FIG. 9 is a diagram showing in full lines how an ordinary size spectaclelens may be cut out of the large lens formed according to thisinvention, and in dot-dash lines showing how a lens to fit a half-eyeframe (well known in the art) can be formed substantially wholly belowthe great arc;

FIG. 10 is a side elevational view of lens forming apparatus associatedwith the first embodiment;

FIG. 11 is an end view of the same taken at the right-hand end of FIG.10;

FIG. 12 is a partial end view taken along the line 12--12 of FIG. 10;

FIG. 13 is an end view taken from an intermediate position along theline 13--13 of FIG. 10;

FIG. 14 is a side elevational view of the right-hand portion of FIG. 10showing another position of the parts;

FIG. 15 is an edge view, enlarged, of the cam follower and the abradingtool in the apparatus of FIG. 10, in which said tool has a sharpabrading edge in the form of a circle;

FIG. 16 is an edge view showing a modification of the cam follower andabrading tool of the apparatus of FIG. 10, in which the edge of the camfollower or tool is a circular arc in section, and the edge may becalled toric;

FIG. 17 is a side elevational view of apparatus associated with thesecond embodiment of this invention;

FIG. 18 is a side elevational view of the work holder at the left end ofFIG. 17 showing one position of the parts;

FIG. 19 is an end view taken on the line 19--19 of FIG. 17;

FIG. 20 is a view similar to FIG. 18 showing a modified arrangement forpreventing slippage between the evolute cam and its coacting verticalplane;

FIG. 21 is a view, enlarged, taken along the line 21--21 of FIG. 17;

FIG. 22 is an end view taken at the left end of FIG. 17; while

FIG. 23 is an end view taken at the left end of FIG. 17 showing adifferent position of the parts.

Several novel features of the lens of this invention, illustrated inFIGS. 1 through 5, differentiate it from existing ophthalmic lenseswhich change in power to correct presbyopia. These features aredependent upon the geometry of the unique front surface, hereinaftercalled the variable surface, at least a portion of which changes inoptical power in a continuous and regular manner to provide the addedrefractive power necessary to correct the accommodative insufficiency orabsence of accommodation in presbyopia and in aphakia. These novelfeatures of the variable surface are:

1. A specific axis at the intersection of a pair of orthogonal principalplanes each of which intersects the variable surface normally at allpoints, the two planes meeting on the surface at an umbilical pointhereinafter called the axial point.

2. The first of said planes, generally horizontal, which is thehorizontal principal plane, intersects said variable surface in agenerally horizontal circular or elliptical arc, hereinafter called thegreat arc, with the prolate point of said arc, when elliptical,coinciding with the axial point, said horizontal principal plane being areal or potential plane of symmetry, as will be discussed later.

3. The second of said planes, generally vertical in direction, which isthe vertical principal plane, is a plane of symmetry which intersectssaid variable surface in a curved line hereinafter called the principalcurve, the portion of said principal curve below the axial point beingelliptical with its oblate point at the axial point and increasing incurvature in an accelerated manner while the portion above said axialpoint is either circular, or elliptical with its prolate point at theaxial point and identical to either half of the great arc, or is similarin shape to that portion below the great arc and with its oblate pointat the axial point. When the portion above the great arc is the sameshape as that below, the horizontal principal plane is a plane ofsymmetry.

4. At the axial point, the derivative of curvature of the variablesurface vanishes.

5. Along the principal curve, from the axial point downward, thecurvature of the variable surface increases continuously and regularlyin an accelerated manner, and at all points along said portion of theprincipal curve, in directions orthogonal to it, the curvature of saidsurface also increases in a continuous and regular manner from the axialpoint downward at an accelerated rate substantially equal to that alongthe principal curve itself. From the great arc upward, the surface maybe spherical, or a prolate ellipsoid or slight modification thereof, orsimilar to, or identical to that portion below the great arc, as will bedescribed later.

6. Along the principal curve from the axial point downward, and from theaxial point upward when the portion of the principal curve above thegreat arc is not circular, all plane sections of the surface orthogonalto the principal curve, hereinafter called transverse sections, areconics of eccentricity greater than zero (in the broad sense hereinabovereferred to) whose axes containing both foci lying in the verticalprincipal plane intersect normally the principal curve. Below the greatarc, the curvatures at the apices of consecutive conic transversesections at the principal curve increase at an accelerated rate down theprincipal curve from said great arc at a rate equal to that along saidportion of the principal curve while the corresponding eccentricities ofsaid conics also increase. When the portion of the principal curve abovethe great arc is elliptical with its oblate point at the axial point,the curvature at the apices of consecutive conic transverse sections atthe principal curve increase at an accelerated rate up the principalcurve from said great arc at a rate substantially equal to that alongsaid portion of the principal curve while the correspondingeccentricities of said conics also increase. When the portion of theprincipal curve above the great arc is elliptical with its prolate pointat the axial point, the curvature at the apices of consecutive conictransverse sections at the principal curve decrease at an acceleratedrate up the principal curve from said great arc at a rate less than thatalong said portion of the principal curve while the correspondingeccentricities of said conic transverse sections also decrease. When theportion of the principal curve above the great arc is circular, alltransverse sections of the variable surface above the great arc arecircular and identical.

7. The variable surface is unique in that in addition to having agenerally vertical principal curve of varying curvature, at least inthat portion of the principal curve below the axial point, it also has agreat arc which is either a circular umbilical arc along which thederivative of curvature of the variable surface vanishes, see FIG. 4,and hence said variable surface along such great arc can be osculated bya matching sphere, or an elliptical arc along which the derivative ofcurvature at said arc of all sections orthogonal to it vanish, see FIG.5, and hence said variable surface along such great arc can be osculatedby a matching prolate ellipsoid of revolution. The fact that thederivative of curvature at said great arc, either circular orelliptical, of all sections orthogonal to it vanish, is fundamental andmakes it possible for the variable surface to be composed of twodistinct portions, a portion above and a portion below the great arc,and yet be perfectly continuous and regular across said great arc,without geometrical or optical discontinuity. Hence, the portion of thevariable surface above the great arc can be a spherical surface or aprolate ellipsoid surface, each a surface of revolution, and yet eachcan be geometrically and optically continuous with the portion below thegreat arc which is not a surface of revolution. Hereinafter when Idescribe the portion of the variable surface below the great arc, it isto be understood that the same description may apply to the portionabove the great arc which may be similar to or identical to the portionbelow.

In FIG. 2, I have shown the geometry of one embodiment of the uniquevariable surface of the lens of this invention as outlined in the sevenprevious points. Arc QBP is the principal curve; QB is circular and BPis elliptical with OB being the sem-minor axis and OA the semi-majoraxis of the ellipse providing elliptical arc BP. Arc WBV is the greatarc. The vertical principal plane contains the principal curve and theaxis of the surface ZZ'. The horizontal principal plane contains thegreat arc and axis ZZ'. Point B is the axial point on the variablesurface outlined by the circle WQVP. Arcs LBL', LTL', LSL' and LRL' arecircular, all of radius length GB, and each of said arcs across thevariable surface represents a circular transverse section. Line LGL'represents a diameter of the sphere contributing the spherical portionof the variable surface WQVP. Arcs EFE', KMK' and NHN' are conics, eachof said conics representing a transverse section across the variablesurface and each of said conic transverse sections, in the order given,being greater in apical curvature and in eccentricity than the precedingtransverse section, and the apical curvature of each transverse sectionbeing substantially equal to the corresponding vertical curvature alongthe principal curve, with both said transverse and vertical curvaturesincreasing at an accelerated rate along elliptical arc BP. Arc GC is theevolute of elliptical arc BA as well as the locus of centers ofcurvature of the apices of the corresponding conic transverse sections.As an example of the centers of curvature at a point along the principalcurve, consider point M. A normal from point M is tangent to evolute GCat point U, the center of both vertical and transverse curvatures atpoint M.

FIG. 3 is used to illustrate various embodiments of the lens of thisinvention in which the upper portion of the variable surface differs ineach of the embodiments being either spherical; or a prolate ellipsoid;or identical to; or similar to; the lower portion, and yet retaining theessential features characteristic of the lens of this invention, namely,a lens having a variable surface with orthogonal principal planes whichintersect said variable surface normally at all points in a great arcand a principal curve, axis, axial umbilical point at which thederivative of curvature vanishes, geometrical and optical continuityacross said great arc, and a continuous and regular increase inrefractive power from the great arc down said variable surface whereinthe transverse curvatures and substantially equal to the correspondingvertical curvatures along said principal curve.

The lens of this invention is made of transparent homogeneous opticalmaterial, either glass or plastic, with glass preferred for spectaclelenses. When finished as an ophthalmic spectacle lens, it will have theusual appearance of an ophthalmic lens, i.e., it will be shaped toconform to a spectacle frame and will be of usual thickness. Whenfinished as a corneal contact lens of ophthalmic plastic it will havethe usual appearance of a corneal contact lens, approximately the sizeof the cornea or slightly smaller, ballasted (thickened) at the bottomof the lens to maintain the principal curve substantially vertical.Hereinafter, the description of this invention will be with reference toophthalmic spectacle lenses, it being understood that corneal contactlenses embodying the principles taught in this invention are included aspart of this invention.

The variable surface is preferably the front convex surface of the lens,although optically equivalent properties may be obtained when theprinciples of this invention are incorporated in an appropriately curvedvariable surface which is posterior and concave.

Ophthalmic spectacle lenses are generally supplied by lens manufacturersto prescription shops in two forms: First, as an excessively large andthick lens having one of its surfaces finished to requiredspecifications. This lens is then modified by generating and polishingthe opposite surface so as to incorporate a patient's prescription andreduce lens thickness to a desired amount, and further modified byedging the lens to the desired shape; and second, as a lens opticallyfinished on both surfaces and of desired thickness. This lens is thenedged to the desired shape by the prescription shop. The first form iscalled a semi-finished lens while the second form is called a finisheduncut lens. In this specification, only the semi-finished form of thelens of this invention will be described in detail, it being understoodthat semi-finished ophthalmic lenses are further processed byprescription shops to a finished lens of ordinary thickness. Thefinished uncut lens produced by the manufacturer is the same as asemi-finished lens which has been generated, ground and polished by aprescription shop, but not edged to shape.

In a manner analogous to that for conventional bifocals and trifocals,for the non-aphakia patient, for the purpose of minimizing aberrations,the most widely used embodiment of the lens of this invention isdesignated as +2.25, +4.25, +6.25, +8.25 and +10.25 base curves, thedesignation of a base curve being the nominal power of the variablesurface at its axial point. For each base curve, there are a series ofadds, the add being the difference in dioptric powers of the variablesurface between that at the axial point and that at a predetermineddistance down the principal curve, 25 mm below the axial point forexample.

The same rules of coflexure for minimizing aberrations which apply tospherical lenses and toric lenses, well known in the art as correctedcurve lenses, can also apply to the lens of this invention in thehorizontal principal plane along the great arc. For lens powers fromabout -20.00 diopters to about +7.50 diopters through said axial point,the great arc can be circular, and the lens of this invention with saidcircular great arc can be considered equivalent to a corrected curvespherical or toric lens through said great arc. Hereinafter I will speakof such lenses as the "corrected curve lenses of this invention". Forlens powers above +7.50 diopters through said axial point, the great arccan be elliptical with the prolate point of said elliptical great arc atthe axial point, and said lens of this invention can be considered,along said great arc, equivalent to a strong plus aspheric ophthalmiclens designed to correct oblique power error and oblique astigmaticerror, such strong plus lenses being designed and used primarily for thecorrection of aphakia, the coflexure of said strong lenses beingdesignated as that at its axis, or axial coflexure. It should beunderstood, however, that the lens of this invention, for powers below+7.50 diopters, need not follow the conventional base curvespecifications and conventional coflexure in the horizontal principalplane, but may be made with an elliptical great arc so that thecorrection of oblique power error and oblique astigmatic error alongsaid great arc, is a function of the changing curvature along said greatarc as well as the axial coflexure.

In order to achieve the variable surface of the lens of this invention,it is essential that the derivative of curvature of the principal curvevanish at the axial point and that at said axial point the upper andlower portions of the principal curve be tangential and of equalcurvature. With these criteria the principal curve, at least below theaxial point, can be an elliptical arc with its oblate point at the axialpoint, or a cycloid with its apex at the axial point, or slightmodifications of said curves or other similar curves, or generallystated, an involute arc in which at its origin at the axial point, thederivative of curvature vanishes. Since the derivative of curvature of acircle is zero, a circular arc whose curvature is that of the variablesurface at its axis may be used as the upper portion of the principalcurve which is continuous at said axial point with said lower portionwhich is either an elliptical arc, a cycloid, or modifications of suchcurves as described above, the portion of the variable surface above thegreat arc being spherical. The derivative of curvature of the ellipticalarc at its prolate point also vanishes so that the principal curve canbe formed of a lower portion elliptical arc joined at its oblate pointto another elliptical arc at its prolate point as the upper portion, andsuch a principal curve can be used in the variable surface of the lensof this invention for the cataract range of lenses. Other principalcurves can be designed which meet the criteria stated above.

The elliptical arc may serve as all or part, or parts, of the principalcurve of the variable surface of the lens of this invention and it willbe used as an example to illustrate how a principal curve of thevariable surface can be constructed, it being understood that this is byway of illustration only and it is not intended to limit this inventionto principal curves of elliptical arcs only.

The first example will be that in which an upper portion circular arc isjoined to a lower portion elliptical arc at its oblate point, bothportions of said curve having the same radius of curvature at theirjunction which is at the axial point of the variable surface. Such aprincipal curve is used in a corrected curve lens of this inventionhaving a variable surface with an upper spherical portion and in whichthe great arc is circular, coflexure being used to correct aberrationsthrough said upper portion. The second example will be that in which anelliptical arc as the upper half of said principal curve is joined atits prolate point to a second elliptical arc at its oblate point andserving as the lower half of said principal curve, said prolate and saidoblate points being of equal curvature at the axial point. Such aprincipal curve is used in the variable surface having an upper portionwhich is an ellipsoid of revolution and in which the great arc iselliptical, the appropriate ellipsoid of revolution portion incombination with a given spherical or toric posterior surface being usedto correct aberrations through said upper portion.

First consider an elliptical arc to be used as the lower portion of theprincipal curve. By suitable choice of an ellipse whose arc is utilizedas said lower part, the radius of curvature at said axial point and theradius of curvature at a given point along said elliptical arc can bemade equal to predetermined values.

In FIG. 6, I have drawn elliptical arc A'BA with A'O and AO thesemi-major axes of the ellipse and with OB the semi-minor axis aboutwhich arc A'BA is symmetrical. Point B is the axial point referred topreviously. Point O is the center of the ellipse. OG is an extension ofsemi-minor axis OB. Arc C'G is that branch of the evolute of the ellipsecorresponding to elliptical arc segment A'B, and GC is that branch ofthe evolute corresponding to elliptical arc segment BA.

Since the two halves of FIG. 6 are symmetrical about BOG, only the righthalf will be considered in the following discussion. Using Cartesiancoordinates with the point O as origin, where a is the coordinate of theellipse in the direction of the semi-major axis, OA, of length A, theabscissa, and b is the coordinate of the ellipse in the direction of thesemi-minor axis, OB, of length B, the ordinate, the radius of curvature,r(a,b), for any point P(a,b), along arc segment BA is given by theequation: ##EQU2## By setting a equal to zero, equation (3) becomes:

    r(axial) = A.sup.2 /B                                      (4)

let r(axial) be a predetermined radius of curvature at said axial point,for example the radius of curvature of one of the base curve valuespreviously mentioned. The value r(a,b) can represent a predeterminedradius of curvature for a predetermined value a such that the dioptricpower at point P(a,b) for the optical material used, is greater than thedioptric power at the axial point by a desired amount, 1.25 diopters forexample, (where dioptric power is given by the well known equation, D =(n - 1)/r, n being the index of refraction of the optical material).

The value of b is determined by the following equation:

    b = B(1 - a.sup.2 /A.sup.2).sup.1/2                        (5)

an ellipse can be completely defined by two appropriate numbers, thelengths of the semi-major and the semi-minor axes, for example, or thelength of the focus, f, and the eccentricity, e. Given the radii ofcurvature of two specific points on the ellipse, one of said pointsbeing at the minor axis and the other at a known distance from the minoraxis, and using equations (3), (4) and (5), the ellipse may be readilydefined.

Rewriting equation (4) as:

    A.sup.2 = r(axial)B                                        (6)

and using the value for b from equation (5) and the value of A² fromequation (6), equation (3) can be simplied and rewritten as: ##EQU3##Since r(a,b), r(axial) and a are predetermined values, the value B (theonly unknown in equation (7) can be solved for. Rewriting equation (7)in the following form to solve for B: ##EQU4## the value of B obtainedwith equation (8) is then used in equation (4) to obtain the value of A.Hence A and B are known and the ellipse is defined. To express theellipse in terms of the parameters e and f, the following equations areused:

    e = (1 - B.sup.2 /A.sup.2).sup.1/2                         (9)

and

    f = (1 - e)A                                               (10)

as a specific example of an elliptical arc appropriate for the lowerportion of the principal curve of the variable surface of the correctedcurve lens of this invention (the first example), let the dioptric powerat the axial point be +4.15 diopters (4.25 diopter base curve), and letthe desired vertical power at a = 0.0250 meters be +5.40 diopters. Letthe refractive material be crown glass of n = 1.5230. Then r(axial) =0.126024 meters and r(a,b) = 0.0968519 meters.

Applying these values to equation (8), the value of B determined is B =0.0247550 meters and by means of equation (4), A = 0.0558545 meters. Bymeans of equations (9) and (10), e = 0.896420 and f = 0.00578540 meters.Thus the portion of the principal curve below the axial point in theform of an elliptical arc is determined. Hereinafter when I refer to thespecific example, it will be the ellliptical arc hereinabove determinedwhich will be considered.

using equation (5) and the pair of values A and B previously determinedfor the specific example, values of b are computed for a series ofvalues of a ranging from 0.0000 to 0.0350 meters, a increasing in stepsof 0.0001 meters, or less if desired. The sets of values A, B, a and bare then utilized in the following equation to determine the dioptricpower D(a,b)(vertical) for the series of points P(a,b) along theprincipal curve: ##EQU5## with A, B, a, and b expressed in meters.

The rate of change in dioptric power, D'(a,b)(vertical), with respect todistance s along the principal curve, or dD(a,b)/ds,

is given by the following equation: ##EQU6## To obtain D' indiopters/cm, the value obtained by equation (12) must be multiplied by10⁻ ².

Note that if either a or b in equation (12) goes to zero, the derivativeof curvature vanishes. Hence with either (or both) a prolate point or anoblate point at the axial umbilical point, the derivative of curvatureof said surface at said point vanishes.

I have described the vertical and transverse powers along the lowerportion of the principal curve as being substantially equal. I intendincluding within the definition of substantially equal not only thecondition of exact equalty of said vertical and transverse powers, butthose small predetermined systematic differences which can be useful forthe correction of oblique astigmatic error for vision through the lensof this invention along said lower portion of the principal curve. Theexpression relating D(a,b)(transverse) and D(a,b)(vertical) whichincludes said systematic differences is:

    D(a,b)(transverse) = D(a,b)(vertical ± ΔD'(a,b)(vertical) (13)

where Δ is a value between 0.0 and 0.2 and D'(a,b)(vertical) is themagnitude in diopters taken from the value D' expressed as diopters/cm.

The radius of curvature, r(a,b)(transverse), of each of said transversesections along said principal curve determined by equation (13) is:##EQU7##

In order to simplify the description of this invention, the examplesused in this specification will be that in which D(a,b)(transverse) andD(a,b)(vertical) are equal along said lower portion of the principalcurve and the variable surface has no astigmatism along said portion ofthe principal curve.

If the transverse sections were circular, there would be an increasingastigmatism lateral to said lower portion of the principal curve,hereinafter called lateral astigmatism. The amount of said lateralastigmatism, V, in diopters, for any point on the variable surface at adistance h, in centimeters, lateral to the principal curve would equal htimes twice the rate of change in refractive power in diopters percentimeter along the principal curve at the level of said point.Expressed as an equation: ##EQU8## The principal directions of saidlateral astigmatism would be approximately at 45° and 135° and saidastigmatism would cause an increasing blurring and distortion for visionthrough increasingly lateral and lower portions of said lens below thelevel of the great arc.

A novel and important feature of the lens of this invention whichreduces distortion and lateral astigmatism to considerably less thanthat predicted by equation (15) resides in the unique design of thevariable surface of the lens in which the transverse sections are conicswhich increase progressively in eccentricity from the great arcdownward. At increasingly lower portions of the variable surface wherethe rate of increase in refractive power along the principal curve ishigh, the eccentricity of the conic transverse sections is also large.The rate of curvature decreases from the apex of a conic laterally,increases with the eccentricity of the conic, hence in the lower portionof the variable surface where the rate of curvature increase down theprincipal curve is large, the decrease in curvature along the transversesections is also large. The effect of said conic transverse sections ascompared to that of circular transverse sections is a reduction in thedownward and laterally increasing curvature of the variable surface onboth sides of the principal curve with the result that both lateralastigmatism and distortion are relatively small in magnitude.

In Table I, I have listed for the specific example some of the values ofa, ranging from 0.0000 to 0.0350, and the corresponding values ofD(a,b)(vertical), D'(a,b)(vertical, D(a,b)(transverse), r(a,b)(verticaland r(a,b)(transverse).

                  Table 1                                                         ______________________________________                                        a      D(vert)  D'(vert) D(trans)                                                                             r(vert) r(trans)                              Meters Diopters Diopt.cm Diopters                                                                             Meters  Meters                                ______________________________________                                        0.0000 4.1500   0.0000   4.1500 0.126024                                                                              0.126024                              0.0050 4.1904   0.1628   4.1904 0.124809                                                                              0.124809                              0.0100 4.3157   0.3412   4.3157 0.121186                                                                              0.121186                              0.0150 4.5388   0.5542   4.5388 0.115229                                                                              0.115229                              0.0200 4.8852   0.8298   4.8852 0.107058                                                                              0.107058                              0.0250 5.4000   1.2138   5.4000 0.096852                                                                              0.096852                              0.0300 6.1638   1.7902   6.1638 0.084850                                                                              0.084850                              0.0350 7.3285   2.7277   7.3285 0.071365                                                                              0.071365                              ______________________________________                                    

Through each point P(a,b) along said elliptical portion of the principalcurve below the axial point, a normal to said curve intersects the majoraxis of the ellipse of said elliptical portion at a distance ae² fromthe center O of said ellipse and continues to the point of tangency onthe evolute of said elliptical portion of the principal curve at thepoint P(α,β) which is the center of curvature of the infinitesimallength of arc about point P(a,b). The slope θ of said normal withrespect to said major axis is: ##EQU9## The coordinates of P(α,β) are:

    α = a - r(a,b) cos θ, and                      (17)

β = b - r(a,b) sin θ (18)

Hence, for every point P(a,b) along said elliptical portion of theprincipal curve, there is a corresponding angle θ and a correspondingpoint P(α, β) on said evolute. See FIG. 6.

It should be noted that the distance GP(α ,β ) along the evolute plusthe distance r(a,b) which is the same as the distance P(α,β) P(a,b), isa constant equal in magnitude to GB, the radius of curvature r(axial).Therefore, if the evolute GC of FIG. 6 were rotated counterclockwise androlled along a fixed vertical line GB without slipping, then all pointsP(a,b) of said elliptical portion of the principal curve would passthrough point B, with said principal curve always being perpendicular tofixed vertical line GB. It is this fact which provides the basis forobtaining the desired conic portion of the principal curve of thevariable surface of the lens of this invention.

Using equations (16), (17) and (18) and the values of a, b, and epreviously determined for the specific example, a sequence of values ofθ, α and β are calculated for each value of a ranging from 0.0000 to0.0350 in steps of 0.0001 meters. The series of coordinates, α and β,are used in the machining of a rolling evolute cam, hereinafter calledthe evolute cam, for the apparatus of this invention.

In Table 2, I have listed for the specific example some of the values ofa, α, β, θ and γ, where γ = (90 - θ). For each value of a there is aspecific rotation γ of the evolute cam which is necessary for theproduction of said desired elliptical portion of the principal curve ofthe variable surface below said axial point.

                  Table 2                                                         ______________________________________                                        a       α    β    θ γ=(90- θ)                    Meters Meters     Meters     Degrees Degrees                                  ______________________________________                                        0.0000 0.000000   -0.101269  90.0000 0.0000                                   0.0050 0.0000322  -0.100055  87.7188 2.2812                                   0.0100 0.000258   -0.096439  85.3889 4.6111                                   0.0150 0.000869   -0.090513  82.9560 7.0440                                   0.0200 0.002061   -0.082431  80.3536 9.6464                                   0.0250 0.004025   -0.072416  77.4922 12.5078                                  0.0300 0.006955   -0.060779  74.2403 15.7597                                  0.0350 0.011044   -0.047932  70.3855 19.6145                                  ______________________________________                                    

If a right circular cone is sectioned by a plane, the sections obtainedare termed conic sections or conics. There are two classes of conics,those of eccentricity less than 1.0, the closed conics or ellipses, andthose of eccentricity greater than 1.0, the open conics or hyperbolas.Separating these two classes of conics is the parabola, of eccentricity1.0, obtained when the sectioning plane is parallel to one of thestraight line elements of the surface of the cone. If the sectioningplane is perpendicular to the axis of the cone, the ellipse obtained ofeccentricity 0.0 is a circle.

Consider now in FIG. 7 a right circular cone with element OL horizontaland in the plane of the drawing. Let ω represent the angle which saidelement makes with the axis OC of the cone, also in the plane of thedrawing. Through a point P along element OL, at a distance l from theapex of the cone, I have drawn lines 1, 2, 3 and 4, representing fourplanes perpendicular to the plane of the page sectioning said cone.Plane 1 is perpendicular to the axis of the cone and sections the conein a circle. Plane 2 is parallel to element OL' and sections the cone ina parabola. Plane 3 is perpendicular to element OL and sections the conein a hyperbola. Plane 4 is parallel to the axis of the cone and sectionsthe cone in a hyperbola of greatest possible eccentricity for said cone.In this example, the conic sections produced by sections between planes1 and 2 are ellipses. The eccentricity e(conic) of the conic sectionproduced by sectioning the cone is given by the following equation:

    e(conic) = sin(ω + φ) sec ω                (19)

where φ is the angle measured from that plane perpendicular to elementOL to the sectioning plane, φ being negative when the sectioning planeis angled clockwise with respect to said plane perpendicular to elementOL and positive when angled counterclockwise with respect to said plane.

The apical radius of curvature of the conic is given by the followingequation:

    r(apical) = l tan ω|cos φ |    (20)

In the production of the variable surface of the lens of this invention,I used in the apparatus a right circular cone as a cam, hereinaftercalled the cone cam, for the production of all transverse sections ofthe variable surface, of eccentricity 0.0 and greater. A circular camfollower, capable of motion parallel only to a single vertical planeperpendicular to the horizontal axis of said cam follower andperpendicular to the vertical plane containing the axis of the cone camand element OL, is made to roll across element OL in a predeterminedsequence of distances l and a corresponding predetermined sequence ofangles φ, both positive and negative, to move in circular, elliptical,parabolic and hyperbolic arcs. Simultaneously, with each adjustment ofthe predetermined sequence of adjustments of the cone cam, there is anassociated adjustment of a predetermined sequence of incrementalrotations of the evolute cam to which a work holder and lens workpiecehave been attached. A rotating circular diamond edged generating toolmoving across the workpiece in pathways parallel to said circular camfollower and on a common axis, the generating tool and the cam followerbeing of equal diameter, forms the basis of the apparatus and method ofproduction of the variable surface of the lens of this invention.

In FIG. 10, line GB is the base line of vertical face Q of plate L alongwhich evolute cam face EC, being the evolute of the principal curve tobe formed on the lens, of cam R rolls vertically upward as it is rotatedcounterclockwise in the plane of the drawing. Cam R is controlled by rod16 passing through slide 11 reciprocatably in way block 110 by actuator10, threaded in fixed block 111 and fixed in slide 11 but rotatablerelative thereto. The adjustment is by knob 21. A plane, perpendicularto the plane of the drawing, containing the face Q and line GB, ishereinafter called the generating plane. Attached to plate L by a pairof bolts or screws, not shown, is a sliding cam SC which is exactlycalculated to support, at the proper height, the lower edge of evolutecam face EC as said evolute cam rotates without sliding up vertical faceQ. Horizontal bar LB fixed to, and spanning way block 110, supportsplate L and cam SC. The work holder WH attached rigidly to arm D whichextends rigidly but adjustably from evolute cam R, has affixed to it bymeans of pitch or other adhesive or mechanical means, the workpiece WPwhich consists of a lens blank 70 mm in diameter, or other suitablediameters, with its upper or exposed surface curve approximately likethe desired finished variable surface. The position of the work holderand workpiece is such that the lens blank surface is bisected by acentral plane through GB, which plane is perpendicular to the generatingplane. The work holder is attached to arm D by horizontal slide HS andlock screw LH while arm D is attached to the evolute cam by verticalslide VS and lock screw LV. By means of slide VS, the height of theworkpiece is set slightly above the level of point B and locked inposition by lock screw LV. By means of slide HS, the work holder isadjusted to such position on arm D that during the process of generatingthe variable surface, th axial point will be at the geometrical centerof the generated workpiece, i.e., the normal to the principal curvewhich is tangent to the evolute at point G will coincide with the minoraxis of the elliptical arc portion of the princial curve. However, insome cases the work holder is adjusted so that the axial point is atsome other desirable point on the principal curve of the lens.

Attached to platform T are two vertical support arms, SA, each bored atthe upper end horizontally and parallel to the generating plane and eachbore bearing a cylindrical rotatable cam pin on a common horizontal axisZZ', hereinafter called the cam axis, which is parallel to thegenerating plane and at the same level as point B in the generatingplane. Extending downward from and perpendicular to said rotatable campins are parallel cam supports, shaft AA and worm gear Y, both of whichin turn are attached perpendicularly to horizontal cam slide assembly Kwhich is at all times parallel to cam axis ZZ'. Resting on slideways inslide assembly K is a portion of a right circular cone, the cone cam CCwhose uppermost element OL is parallel to slide assembly K, whichelement lies in the plane which contains base line GB and isperpendicular to both the generating plane and the cam axis ZZ', whichplane is hereinafter called the central plane. The central planecontaining element OL and the axis of cone cam CC divides the cone camsymmetrically. By means of the cam pins, slide assembly K is rotatableabout cam axis ZZ' and said angular rotation is accurately adjustable bymeans of worm gear Y and worm W, as indicated by pointer 56 on scale 57,and the amount of angular rotation may also be indicated by means ofpointer P' (on pedestal 55) on radial dial D', or alternatively by meansof a flexible shaft from worm W to a mechanical counter, not shown.

Cone cam CC by means of screws MS is movable along slideways SS, asshown in FIGS. 12 and 14, in directions toward and away from thegenerating plane. The control point CP (FIG. 11) for this movement is atthe intersection of cam axis ZZ' and the central plane and always lieson element OL and at the same level as point B. This adjustment isindicated by pointer 50 on scale 51 or by pointer 52 on scale 53 or theamount of movement may be indicated by a mechanical counter connected toscrew MS by a flexible shaft, not shown.

Resting on cone cam CC is circular cam follower CF which has a sharpcircular edge which lies in a vertical plane which also contains camaxis ZZ', said plane hereinafter called the cam plane. Cam follower CFis attached by linkages to table T such that it can roll across cone camCC, the sharp circular edge of said cam follower always remaining in thecam plane, the axis of said cam follower always remaining perpendicularto the generating plane. A diamond edged generating tool GT in the formof a rotatable circular disc of the same diameter as the cam followerand having a sharp circular edge, is supported by said linkage such thatits abrading edge lies in the generating plane. Said generating tool isperpendicular to, and coaxial with, said cam follower axis and is madeto rotate rapidly about said axis by motor M, attached to the tool byflexible shaft 40.

As stated previously, the variable surface below the great arc has anincreasing eccentricity, e(transverse), of successive transversesections along the principal curve from the axial point downward. Theeccentricity of said conic transverse sections may increase uniformlyalong the principal curve or may increase in an accelerated manner,depending upon the desired lateral effects at given levels down thelens. A very satisfactory design utilizes a uniform rate of increase ineccentricity per unit distance a down the lens. Expressed as adifferential equation:

    (de/da) (transverse) = k                                   (21)

As an example, k may be a value of 0.5 eccentricity units per cm downthe lens or in terms of meters, 50 eccentricity units per meter. Thiswould be an increase of 0.005 eccentricity units for each successiveconic transverse section when a increases in steps of 0.0001 meters.

To calculate the required valves of φ (see FIG. 7) as the adjustments ofthe angle of slide assembly K necessary for the production of thedesired eccentricities e(transverse) of the conic transverse sections,for successive values of a, equation (19) can be rewritten as follows:

    φ = sin.sup.-.sup.1 [e(transverse) cos ω] - ω(22)

where e(transverse) replaces e(conic) of equation (19).

Having determined the required values of φ and having determined thevalues of r(a,b)(transverse) along said lower portion of the principalcurve, the values of l may be determined from equation (20) rewritten asfollows: ##EQU10## where r(apical) = r(a,b)(transverse). Thus therequired adjustments of φ and l of the cone cam are determined for theproduction of the desired transverse sections of the variable surface.

In the generation of the variable surface of the lens of this inventionhaving a spherical portion above the great arc, the cone cam is adjustedto the values φ = -ω and l = r(apical)/sin ω

where r(apical) is equal to the value r(axial).

Starting with the cone cam set at the values as above determined, andthe evolute cam pivoted about a horizontal line through point G in thegenerating plane, at the intersection of vertical face Q and the bearingsurface of sliding cam SC, and such that the edge of the workpiecefarthest from the cone cam is just at the generating plane, the camfollower is made to roll back and forth across the cone cam while thegenerating tool GT is rotating rapidly.

The apparatus for rolling the cam follower CF and the generating tool GTback and forth across the workpiece is best understood from FIGS. 10 and11 of the drawings. It is sufficient if such apparatus permits travel ofthe generating tool and the cam follower completely across the work. Forexample, mounted rigidly on the table T are two clevis blocks 23 and 24in which are oscillatably mounted two generally vertical arms 25 and 26which rotate about horizontally aligned pins 27 and 28. The upper endsof arms 25 and 26 are pivotally connected through bearings 25', 25' witha unitary cross rod 29. Collars 26', 26' hold rod 29 in place. Parallel,generally horizontal arms 30 and 31 are pivotally mounted on cross rod29 by bearings 30' and 31'. On the ends of arms 30 and 31 most remotefrom the vertical arms 25 and 26 are fixedly mounted at right angles twohorizontally spaced pairs of support arms 32 and 33. The arms 30, 31, 32and 33 are rigidly held together by horizontal plate 34 and verticalplate 35 as well as by bolts 36 and 37. At the lower ends of the supportarms 32, a shaft 38 is rotatably mounted in bearings 39 and thegenerating tool GT is fixedly mounted concentrially on shaft 38 betweenthe arms 32. This shaft is rotated by means of motor M, having a fixedmounting, not shown, through a flexible drive cable 40. At the lower endof the arms 33, a short shaft 41 is rotatably mounted in bearings 42,and fixed on shaft 41 is the cam follower CF in such a position that itis axially aligned with the shaft 38. It should be understood that thegenerating tool GT and the cam follower CF are of the same diameter andthe the edge shape may be as shown in FIG. 15 or FIG. 16. It should beunderstood in FIG. 11 that the generating tool GT is directly behind thecam follower CF and arm 25 is directly behind arm 26.

The slide assembly K is best understood from FIGS. 10, 11 and 12.Rigidly mounted on the table T are a pair of blocks 43 and 44 alignedperpendicular to the axis of the clevis pins 27 and 28. To each of theseblocks a support arm SA is rigidly attached in a vertical position. Atthe upper ends of these support arms, pivotally connected by pins 45 and45', are vertical arm AA and worm gear Y which are parallel to the armsSA. Rigidly connecting the lower ends of the arm AA and worm gear Y areparallel generally vertical spacing bars 46 which are rigidly connected,centrally, by a way block 47, generally U-shape in section. End plates47a are fixed at each end. The cone cam CC has a dove tail downwardextension 48a which is complementary to, and a sliding fit in, the slideways SS in the block 47. The cone cam CC has a further downwardprojection 48b in the form of a downwardly extending tongue, throughwhich extends a female thread 49 extending longitudinally of the conecam. The threaded screw MS, parallel to cone cam element OL, isrotatably mounted in the members 47a and passes through and cooperateswith the thread 49 when manipulated by the knob 22 at the right-hand endof the screw MS as clearly seen in FIG. 10. The uppermost element ofsaid cone cam, as marked OL in FIGS. 10 and 12, corresponds to thestraight line element OL of the cone which was previously described inconnection with FIG. 7. The manipulation of the knob 22 and the actionof the threaded screw MS in the female thread 49 serves to adjust thedimension l as indicated in FIG. 7. The position of the cone cam CC andthe extending tongue 48b of the same as indicated in FIG. 14 shows howthese parts have moved toward the right from the position shown in FIG.10. This position is indicated by the pointer 50 which is mounted on theblock 47 and reads on a scale 51 carried by the cone cam CC. There isalso an indication of this movement by the pointer 52 which reads on ascale 53 carried by the knob 22.

The worm gear y serves to rotate the slide assembly K about the axis ZZin a manner illustrated in FIG. 14. The worm gear Y is rigidly attachedto the bar 46 of the slide assembly K as shown in FIG. 12. This gear issubstantially semi-circular in extent. The worm W rotates in block W'fixed to one of the supports SA as seen in FIG. 12 and is part of shaft54. The graduated knob D' serves to turn the shaft 54 and the worm W andsuch adjustment is indicated on the knob D' by the pointer P'. Suchadjustment is also read by the pointer 56 rotatable with pin 45' andreading on scale 57 which is mounted on one of the support arms SA. Thisadjustment sets the value of the dimension φ indicated in FIG. 7.

At the end of each oscillation of the cam follower CF, screw 10,attached to slide 11, is rotated a predetermined amount by knob 21.Sleeve 12, affixed to slide 11 by collar 13, pins 14 and 14' and supportblocks 15 and 15', is freely rotatable about the axis of pins 14 and 14'which is perpendicular to the central plane. See FIG. 13. Shaft 16,snugly and slidably fit within sleeve 12, is caused to move with slide11 and to rotate evolute cam R and cam face EC, which is held firmlywithout sliding against vertical face Q by springs S and S', apredetermined angular increment, indicated by pointer P, attached to pin14, on scale 17, and more exactly by calibrated linear scale 17',attached to slide 11, thereby advancing the surface of the workpiecethrough the generating plane. For the specific example, some of the datafor which is indicated by the asterick in Table 3, an incremental amountof angular rotation of the evolute cam of 0.045463° advances the surfaceof the workpiece through the generating plane an amount of 0.0001 metersas measured along the upper portion of the principal curve. Suchoscillations and angular rotations of the evolute cam are repeated untilthe first or upper portion of the variable surface above the great arcis generated as a spherical portion, a total angular rotation of15.9124°. Thereafter to generate the second or lower portion of thevariable surface below the great arc, at the end of each succeedingoscillation of the cam follower, the cone came is adjusted by means ofknobs D' and 22 respectively to the succeeding values of φ and l toproduce the succeeding values of r(transverse) and e(transverse), whilethe evolute cam is adjusted for each oscillation to advance theworkpiece through the principal plane as said evolute cam rolls upvertical face Q, supported by sliding cam SC, until the second portionof the variable surface is completely generated.

In this operation, the oscillation of frame 25, 26, 29 (carrying the camfollower CF and generating tool GT) is caused by actuating handle 29abut it is understood that this might be motorized.

An alternative procedure for producing the variable surface in which theupper portion is spherical is to start with a preground lens blankworkpiece having a central axis and a convex spherical surface of thedesired radius of curvature, +4.15 diopters, for example. The workpiecewith its convex spherical surface facing upward is fixed to the workholder such that its axis can also be the axis of the variable surfaceat the completion of the generating. Evolute cam R is rotated initiallythe full 15.9124° so that the center of the workpiece convex uppersurface is at point B and the axis of said blank passing through B andnormal to said convex surface coincides with line GB. The height of theworkpiece is adjusted so that the diamond edged generating tool, GT,just contacts the workpiece. The generating process then proceeds asdescribed above for the completion of the lower portion of the variablesurface.

In Table 3, I have listed for the corrected curve lens of this inventionembodying the specific example some of the values of a, ranging from0.0000 to 0.0350 meters, and the corresponding values of r(vertical),r(transverse), γ, φ, and l, and e(transverse), for a 1/2 cone angle of ω= 60° (total angle at apex 2ω = 120°) and a value of (de/da)(transverse)= 0.005 e units per meter, where φ and l have been calculated by meansof equations (22) and (23) for each point P(a,b) along said lowerportion of the principal curve.

As the second example, consider a variable surface for the cataractrange of lenses in which the elliptical arc serving as the upper portionof the principal curve is joined at its prolate point to the ellipticalarc serving as the lower portion of the principal curve, at its oblatepoint. In FIG. 8, drawn to scale, I have shown principal curve QBP withelliptical arc QB joined at its prolate point to elliptical arc BP atits oblate point, the refractive power at the axial point being +14.00diopters for refractive material of crown glass having an index ofrefraction of n = 1.523. The ellipse providing the upper portion of theprincipal curve QB, and the great arc, has an eccentricity of 0.5790.The radius of curvature at the axial point r(axial) is 0.0373571 meters.The semi-major axis of length A' and the semi-minor axis of length B' ofthe ellipse providing the upper portion of the principal curve and ofthe great arc is 0.0561962 meters, and 0.0458148 meters respectively.

                                      Table 3                                     __________________________________________________________________________    a       r(vert)   r(trans)     φ         e(trans)                         Meters  Meters    Meters                                                                              Degrees                                                                              Degrees                                                                              Meters Eccentricity                     __________________________________________________________________________    0.0000  0.126024  0.126024                                                                            0.0000 -60.0000                                                                             0.1455200                                                                            0.0000                           0.0050  0.124809  0.124809                                                                            2.2812 -52.8192                                                                             0.1192360                                                                            0.2500                           0.0100   0.1211186                                                                              0.121186                                                                            4.6111 -45.5225                                                                             0.0998619                                                                            0.5000                           0.0150  0.115229  0.115229                                                                            7.0440 -37.9757                                                                             0.0839416                                                                            0.7500                           0.0200  0.107058  0.107058                                                                            9.6464 -30.3000                                                                             0.0713720                                                                            1.0000                            0.0250*                                                                              0.096852  0.096852                                                                            12.5078                                                                              -28.3178                                                                             0.0600246                                                                            1.2500                           0.0300  0.084850  0.084850                                                                            15.7597                                                                              -11.4096                                                                             0.0499965                                                                            1.5000                           0.0350  0.071365  0.071365                                                                            19.6145                                                                              1.0450 0.0412095                                                                            1.7500                           __________________________________________________________________________

A series of values of a for said upper portion of the principal curvecan be calculated from the following equation:

    a = A'(1 - b.sup.2 /B  .sup.2).sup.1/2                     (24)

for values of b ranging from 0.0000 to 0.0275 meters, b increasing insteps of 0.0001 meters, or less if desired. Semi-finished cataract lensblanks are generally no longer than 5.5 centimeters in diameter so thatthe maximum value of 0.0275 meters for b is adequate.

For each point P(a,b) along the upper portion of said principal curvethere is an angle K between the normal to said curve at P(a,b) and themajor axis of the ellipse providing said portion of the principal curve,where, substituting K for θ in equation (13): ##EQU11## A set of valuesof angle K, can thus be calculated for the corresponding set ofcoordinate values a and b. For each of said points P(a,b) along saidportion of the principal curve about the axial point, the value of theradius of curvature of the elliptical transverse section,r(a,b)(transverse), at its prolate point can be computed from thefollowing equation:

    r(a,b)(transverse) = (b/sin K K)                           (26)

the eccentricity e(transverse) for each of the series of ellipticaltransverse sections of said prolate ellipsoid upper portion of thevariable surface having an eccentricity of e(prolate) is obtained fromthe following equation:

    e(transverse) = e(prolate) cos K                           (27)

to calculate the required values of φ for the adjustments of slideassembly K necessary for the desired values e(transverse) of thetransverse sections of said upper portion of the variable surface,equation (22) is used. Having determined the required values ofe(transverse), and the values of r(a,b)(transverse) obtained by means ofequation (26), the values of l may be determined by means of equation(23), the values of r(a,b)(transverse) being used for r(apical).

r(a,b)(vertical) may be calculated for each of said points P(a,b) bymeans of equation (1), or alternatively, by means of the followingequation: ##EQU12##

The evolute cam in this case is composed of two adjoined geometricalparts; the evolute for the elliptical arc QB used as the upper portionof the principal curve, and the evolute for the elliptical arc BP usedas the lower portion of said principal curve, both evolute portionsbeing continuous at point G of FIG. 8. Arc portion LG is that portion ofevolute G'G related to QB and arc portion GM is that portion of evoluteGC related to BP.

The coordinates for the portion LG of evolute G'G are determined in thesame manner as the coordinates for the portion GM of evolute GC. Thecoordinates a and b for the series of points P(a,b) along the portion ofsaid principal curve above the axial point are used to compute asequence of values α and β by means of equation (17) and (18), K beingsubstituted for φ, and said computed values are used as coordinates formachining that portion of the combined evolute cam corresponding to theupper portion of the principal curve.

In the actual generating process for this cataract lens, the workpiece,curved approximately like the desired variable surface is attached tothe work holder and positioned such that its edge farthest from the camplane is just at point B of the generating plane. As the cam follower CFis made to roll back and forth across the cone cam CC, the generatingtool GT is rotating rapidly and removing lens material. At the end ofeach oscillation of the cam follower, evolute cam R is rotated apredetermined angular increment by turning knob 21 to roll cam face ECup vertical face Q of plate L and advance the workpiece through thegenerating plane. Simultaneously, the cone cam is adjusted arcuatelyfrom its initial setting a predetermined angular increment, calculatedas taught herein, by knob D', and moved along slide assembly K byindicator knob 22, each angular position of evolute cam R for the upperhalf of the variable surface and angular position θ for the lower halfof the variable surface corresponding to the predetermined angularposition φ and linear position l of the cone cam. This process iscontinued until the entire variable surface is generated.

In an alternative procedure for producing the variable surface of thecataract lens of this invention, a preground lens blank workpiece isused in which the upper convex surface with its axis central is aprolate ellipsoid of the desired eccentricity and apical radius ofcurvature. The workpiece with its convex ellipsoid of revolution facingupward is fixed to the workpiece holder such that its axis can be theaxis of the variable surface at the completion of the generation.Evolute cam R is rotated counterclockwise until the apex of theworkpiece surface is at point B and the axis of said surface coincideswith line GB. The height of the workpiece is adjusted so that thediamond edged generating tool GT just contacts the workpiece. The conecam is adjusted to the values of φ and l by means of knobs D' and 22 sothat the generating tool in its oscillation across the workpiece willproduce the desired conic transverse section for the value of γ ofevolute cam R. With each subsequent oscillation of the cam followeracross the cone cam, evolute cam R is rotated to a new setting while thecone cam is likewise reset. This process is continued until the lowerportion of the variable surface is completed. for the

In order to generate the variable surface which symmetry about the axisand about or across each of the principal planes, the generation of thevariable surface commences at the mid-point of the lens as describedabove the lower half of the surface. At the completion of this portionof the variable surface, the work holder and attached workpiece areremoved from arm D, rotated 180°, replaced in the original position onhorizontal slide HS and locked in position by lock screw L. Again thegeneration commences from the mid-point of the lens. At the completionof generation, the variable surface will be symmetrical about the axisof the lens and about both principal planes.

In order to generate a variable surface in which the portions on eitherside of the great arc are similar but not identical, the generation ofthe variable surface commences at the midpoint of the lens as describedabove. At the completion of this portion of the variable surface, thework holder and attached workpiece are removed from arm D. The evolutecam is also removed and replaced with another evolute cam which producesa principal curve having the same r (axial) value as that produced bythe initial evolute cam but which produces an otherwise differentprincipal curve. The work holder and attached workpiece are then rotated180° and replaced on horizontal slide HS and locked in position by lockscrew LH. The height of the workpiece is then adjusted on vertical slideVS and locked in position by lock screw LV. The second half of thevariable surface is then generated in a manner similar to that of thefirst half.

In the description of the invention, I have shown and described the camfollower with a sharp circular edge, see FIG. 15. In actual practice,the sharp circular edge is subject to wear as the cam follower rollsrepeatedly across the cone cam. Instead of a cam follower with a sharpcircular edge, a toric edged cam follower may be used with advantage toproduce the variable surface of the lense of this invention with onlyminimal differences in the shape of the surface produced. Such a camfollower is shown in FIG. 16. The advantage of a toric edged camfollower is primarily its durability compared to that of the sharp edgedcam follower, but it also permits the production of a variable surfacein which the transverse sections are modified slightly from true conicshape and in which the eccentricity of each of said modified conictransverse sections having an apical point of a specific curvaturediffers from that produced by a cam follower with a sharp circular edgeproducing the same apical curvature.

At the completion of the generation of the variable surface of the lensof this invention, marks left by the circular abrading tool consistingof fine pits and scratches along transverse circular and conic arcs mustbe removed by grinding the surface in preparation for polishing. Therepeated cuts of tool GT across the workpiece are so closely spaced thatonly fine grinding and polishing are necessary to finish the variablelens surface without substantial change in its originally formed shape.For this purpose I may use my Lens Grinding Apparatus, described in myU.S. Pat. No. 3,583,111, patented June 8, 1971, When grinding iscomplete and the surface is uniformly smooth, it is polished with ceriumoxide or rouge abrasive in conjunction with a conforming polishing padof felt or Nylon cloth sheet material which permits the polishing pad toconform to the surface contour over a broad area with relatively uniformpressure, while there is continuous relative motion between said surfaceand the polishing pad.

FIG. 9 is a diagrammatic representation of an oblique top view of thesemi-finished corrected curve lens of this invention, partial data forthat portion below the great arc being shown in Table 3, in which thevariable surface WQVP is spherical above the great arc and progressivelyincreasing in curvature below the great arc. Both the great arc and theprincipal curve are marked with a thin line of water-proof ink to aid infinishing the posterior surface in accordance with the patient'sprescription. The posterior surface is left spherical. The thickness ofthe semifinished lens is approximately 8 mm at its thinnest portion toallow for generating, grinding and polishing of the posterior surface bya prescription shop.

Superimposed on the variable surface of FIG. 9 is the outline in fulllines of a possible finished lens area which could represent the frontsurface of a finished ophthalmic lens. Other possible positions may beutilized for said finished lens area, wherein more spherical lens areais available for distance vision and less for near vision, or viceversa. Also a half lens is possible as shown in dot-dash lines.

The separation of 0.0250 meters between the axial point and the addpoint used in the specific example as marked * in Table 3 was used forthe purpose of description, and other separations, 15, 18, 20, 22, 28 or30 mm as example, are included in this invention.

In utilizing the semi-finished lens of this invention for completion asa non-cataract ophthalmic lens, the appropriate base curve semi-finishedlens is first selected. Using the inked great arc and principal curvelines, as shown in FIG. 9, as guides, the variable surface is madeadherent by means of pitch or other adhesive, to a lens block. The lensis positioned on said block so as to utilize the desired portion of thevariable surface for the completed ophthalmic lens and to provide thecorrect principal meridians when a toric surface is to be the backsurface of the lens. The concave surface of the lens is then generated,ground and polished in the usual manner for opthalmic lenses to reducethe lens to normal thickness and to incorporate the correction for thewearer's refractive error. The lens is then edged to the desired sizeand shape to fit a spectacle frame.

The semi-finished cataract lens of this invention is supplied to theprescription shop in terms of the power of the variable surface at theaxial point, there being a large series of such lenses to cover a largerange of prescriptions for aphakia. The add for each lens in such aseries may be set for a specific value, +2.50 diopters, for example, fora given distance below the axial point, 20 mm for example. A similarseries may be produced with the add +3.00 diopters, or +3.50 diopters,etc.

Although for the purpose of this specification the principal curvedescribed and used in the specific example is elliptical below the axialpoint, this invention includes other principal curves having an axialumbilical point where D'(axial) vanishes. Although I have described thetransverse sections below the great arc as conic, this inventionincludes those substantially conic transverse sections which areproduced when the edge of either, or both, the circular cam follower CFand the circular generating tool GT, is toric rather than sharp edged.

A second embodiment of this invention is shown in FIGS. 17 to 23. Whilethe first embodiment provides transverse curves which may be conicswhose eccentricity is less than 1.0 or equal to 1.0, or greater than1.0, this second embodiment can provide transverse curves on the uniquevariable surface which are conics and always elliptical arcs ofeccentricity less than 1.0. For this embodiment a circular cam issubstituted for the cone cam of the first embodiment. However, a verylarge number of useful lenses may be constructed using this secondembodiment.

In FIGS. 17, 18 and 19, the mechanism at the left-hand part of thedrawing, where the tool GT' cuts the workpiece WP' in the work holderWH', is exactly like that described in connection with the firstembodiment although details have been omitted from the drawings. At theright-hand part of the drawing the pattern followed includes a circulararc cam C movable about a vertical radius as its axis to provide variousellipses when projected horizontally toward the left. When the axis ofthe cam follower lies in the central plane, the control point, see FIG.17, is the highest point CP', of the circuit cam C, which lies in thecentral plane. The cam follower CF' is a cylinder so as to maintaincontact with the circular cam pattern in various oriented positions ofthe latter.

The parts associated with the lens generating operation at the left ofFIG. 17 are given the same reference characters as in the firstembodiment with a prime suffix.

If said circular cam C of radius r is rotated about a vertical diameter,between the full line and dot-dash positions of FIG. 17, theperpendicular projection upon the generating plane through G'B' is inellipse whose eccentricity is given by the expression;

    e = sin φ                                              (29)

where φ is the angle of rotation of said circular cam from its originalposition (in full lines). The apical radius of curvature of said ellipser (apical) is also a function of φ, and therefore of e, and is given bythe equivalent expressions:

    r(apical) = r cos.sup.2 φ , and                        (30)

    r(apical) = r(1 - e.sup.2).                                (31)

Thus, the circular cam C capable of being rotated through apredetermined sequence of angular increments about its vertical diameteras an axis can provide the basis for producing a predetermined sequenceof elliptical arcs, each of which increases in apical curvature andeccentricity. To calculate the required sequence of values of positionsφ of said circular cam of radius r for a sequence of values of r(transverse), equation (30) is rewritten as follows: ##EQU13## where thesequence of values of r (transverse) have been determined as previouslydescribed by means of equation (14) and those preceding it. Theeccentricities e of said elliptical transverse sections are obtained bymeans of equation (29), using the values of φ obtained by means ofequation (32).

The use of such an upright circular cam which is rotated incrementallyin a predetermined manner simultaneously with the predeterminedincremental rotation of said evolute cam R' (the pattern for which ismentioned hereinabove) to which a work holder and lens workpiece havebeen attached, forms the basis of the apparatus to produce the variablesurface of the lens of this second embodiment of my invention.

In FIG. 17, line G'B' is the base line of a vertical linear platesurface Q' along which the evolute cam face EC' formed on cam R' rollsvertically as it is rotated counterclockwise in the plane of thedrawing. The right-hand surface Q' of vertical plate L', secured to barLB' and slide 11, and perpendicular to the plane of the drawing,contains line G'B' and is hereinafter called the generating plane. Theadjustment screw 10' with micrometer attachment 21' is attached to slide11' through which passes a sleeve 12' affixed to slide 11' by a collar13', pins on the axis 140 similar to pins 14 and 14', previouslydescribed, and support blocks like those 15 and 15' previously describedin the first embodiment. This makes sleeve 12' freely rotatable aboutthe axis of pins 14 and 14' which is perpendicular to the central plane.Shaft 16', which snugly and slidably fits within the sleeve 12', iscaused to move with slide 11' and to rotate the evolute cam R', which isheld firmly without sliding against the vertical face Q' by springs S1and S2. The cam R' is guided by its lowermost edge at G' traveling alongcam SC'. The angular increment of movement of cam R' is indicated bypointer P', attached to a pin on the axis 140 and is read by protractorscale 17' attached to the slide 11'.

Attached to platform T, at the right-hand side of FIG. 17 is ahorizontal circular table, F, rotatable in recess R about a centralvertical axis which is parallel to line G'B' and in a common plane,hereinafter called the central plane, perpendicular to both thegenerating plane and platform T'. Rigidly attached to horizontal table Fand cam base CB is an upright circular arc cam C of hardened steel platewith a wedge shaped central portion coming to a sharp circular edge ofradius r, hereinafter called the cam circle, a radius of said cam circlecoinciding with said vertical axis, hereinafter termed the circular camaxis. The plane containing the cam circle and circular cam axis istermed the circular cam plane. The highest point CP' of the cam circlewhich is at the circular cam axis in said central plane is at the samelevel as point B' on the base line of vertical face Q'. Resting upon thecircular cam circle is a cylindrical cam follower CF' freely rotatableon shaft 67 supported by ball bearings (not shown), and attached totable T' by a structure including shelf 60, and frame linkages 61, 63,64, 69 which permit said cam follower to roll along said circular camwhile always maintaining the axis of said cam follower perpendicular tothe generating plane. Spring 73 maintains this contact.

These frame linkages are best understood from FIGS. 17 and 23. A shelf60 is fixed to the two parallel side frame members 61 which carryhorizontal shaft 62 and also serves as linkages This shaft is rotatablyconnected by two pairs of parallel links 63 on the left in FIG. 17 and64 on the right in FIG. 17. The lower ends of the links 63 and 64 arerotatably mounted on the shaft 65 which is held at opposite ends in twoblocks 66 secured to table T'. The cam follower CF' is carried by ashaft 67 which is fixedly mounted in the side frame members 61. Thisshaft passes through an open slot 68 in a handle lever 69 operable byhandle 69a. This lever is oscillatably mounted on a pin 70 held in ablock 71 fixed to table T'. The lower end of the lever 69 is held in aclevis in block 71 (not shown) which, on opposite sides of the lower endof lever 69 carries adjustable screw stops 72 (FIG. 17) to limit theoscillation of lever 69. A diamond edged generating tool GT' in the formof a rotatable circular disc, of the same diameter as the cam followerCF', is supported by bearings within bracket sleeve 74 rigidly attachedto linkage 61 such that its abrading edge lies in the generating planethrough G'B'. Said abrading tool is perpendicular to, and coaxial with,said cam follower axis and is made to rotate rapidly on its shaft 75about said axis by belt 76 and shaft 77 supported by bracket 78 anddriven by motor M' supported on shelf 60.

In the generation of the variable surface of the lens of this invention,where the portion above the great arc is spherical, the circular camplane is first set at the zero position in which it is parallel to thegenerating plane, seen in full lines in FIG. 17. The radius of curvatureof the cam circle, r, is equal to r(axial). The work holder attachedrigidly to arm D' which extends from evolute cam R', has affixed to itby means of pitch or other adhesive or mechanical means, the workpieceWP' which consists of a lens blank 70 mm in diameter with the upper orexposed surface curved approximately like the desired finished variablesurface. The position of the work holder WH' and workpiece WP' is suchthat the lens blank surface is bisected by the central plane. The workholder is attached to arm D' by suitable means adjustable like slide HSwhile arm D' is attached to evolute cam R' adjustable like slide VS insuch position that during the process of generating the variablesurface, the axial point will be at the geometrical center of thegenerated workpiece, i.e., the normal to the principal curve which istangent to the evolute at point G' will coincide with the minor axis ofthe elliptical arc portion of the principal curve below the great arc.

Starting with the cam plane at the zero position and the evolute cam R'pivoted about a horizontal line through point G' in the generating planeand such that the edge of the workpiece farthest from the circular camplane is just at the generating plane, the cam follower CF' is made toroll back and forth along the circular cam C while the generating toolGT' is rotating rapidly. At the end of each oscillation of the camfollower, evolute cam R' is rotated by wheel 21' a predetermined amountabout said horizontal line through point G' as previously described inthe first embodiment, thereby advancing the surface of the workpiecethrough the generating plane. Such oscillations and angular camrotations are repeated until that portion of the variable surface abovethe great arc is generated as a spherical portion. Thereafter at the endof each succeeding oscillation of the cam follower, the circular camplane is rotated by wheel 80, worm 81, worm wheel 82 and shaft 85 apredetermined angular increment φ calculated by means of equation (32),read at pointer 83 on indicia 84, while evolute cam R' is also rotated apredetermined amount, and continues to advance the workpiece through theprincipal plane as it rolls without sliding up cam plate surface Q',until the second portion of the variable surface below the great arc isgenerated. Oscillation of shelf 60, frame 61, the cam follower CF' andthe generating tool GT' may be made manually using handle 69a. It isunderstood that this oscillation could be caused by a motor, if desired.

Other than the changes above described, lenses are formed by the secondembodiment the same as in the first embodiment.

At the completion of the generation of the variable surface of the lensof this invention, marks left by the circular abrading tool consistingof the fine pits and scratches along transverse circular and ellipticalarcs, must be removed by fine grinding and polishing as in the case ofthe first embodiment.

Comparing FIG. 20 with FIG. 18, there is shown an alternate way ofpreventing slippage as the evolute cam face EC or EC' rolls on thecoacting vertical cam plate face Q or Q'. In FIG. 20, the vertical faceQ' is provided with a rack 90 and the evolute cam face EC' is providedwith coacting gear teeth 91. Note that no guiding cam surface such asthat shown at SC and SC' is necessary with this construction and thepoint G" coincides with the position G and G' previously described. Thisconstruction for preventing slippage between the evolute cam and itscoacting vertical cam plate may be used in either the first or secondembodiment of this invention.

In the description of both the first and second embodiments of theapparatus and method of this invention, I have shown the coaxial camfollower and generating tool as oscillatory across the cam and workpieceand a fixed center plane. It should be understood that such motions arerelative and that identical effects can be achieved and are includedwithin the scope of this invention with the apparatus constructed suchthat the workpiece, cam and center plane oscillate together in adirection horizontal and perpendicular to said center plane while thecoaxial cam follower and tool are held fixed except for substantiallyvertical oscillation.

Although this specification has been directed primarily at ophthalmiclenses to be used in spectacles, it is also intended to be used forcorneal contact lenses which differ from the spectacle lens only inmagnitude but not in optical principles.

Where there is mentioned herein a curvature at a "point", it should beunderstood as referring to the curvature of an infinitesmally small lineor surface at such point.

The term "cut" as used herein in the specification and claims means"abrade" also.

What is claimed is:
 1. A multifocal ophthalmic lens of homogeneoustransparent optical material, useful for the correction of therefractive error and the accommodative insufficiency or absence ofaccommodation in presbyopia and aphakia, having a geometrically andoptically continuous variable convex front surface at least a portion ofwhich changes continuously and regularly in refractive power, and acoacting coincoid conicold surface of eccentricity zero or greater ortoric back surface, said variable front surface characterized by havinga single pair orthogonal principal planes each of said planes of saidpairs intersecting said variable convex front surface normally at allpoints, the first of said principal planes, generally horizontal in use,intersecting said front surface normally in a conic great arc ofeccentricity zero or greater, the derivative of curvature of saidvariable front surface vanishing at said great arc at least in sectionsby all planes orthogonal to it, said arc providing a unique tangentialjunction between an upper and a lower portion of said variable frontsurface, there being geometrical and optical continuity and a continuousand regular change in curvature and refractive power in crossing saidconic great arc, without localized distortion in the field of visionthrough said lens, the second of said principal planes being a generallyvertical plane of symmetry of said variable front surface, intersectingorthogonally said great arc and said first principal plane and formingthe axis of said variable front surface, said axis intersecting thecenter of curvature of said great arc when its eccentricity is zero andcoinciding with the major axis and intersecting both foci and the centerof curvature of the apex of said great arc when its eccentricity isgreater than zero, and intersecting said variable front surface normallyin a principal curve in which, at the point where said axis intersectssaid variable front surface, namely, the axial umbilical point, thecurvature of said great arc and said principal curve are equal, thederivative of curvature of said principal curve and of said great arc ofsaid variable front surface in all meridian sections containing saidaxis, vanishing or zero at said axial umbilical point, said principalcurve at least below said great arc, increasing in curvature andrefractive power downwardly in a continuous and regular manner and at anaccelerated rate with increasing distance from the axial umbilical pointof said variable front surface, all transverse sections of said variablefront surface below said great arc, by planes orthogonal to saidprincipal curve, being conics of eccentricity greater than zero whoseaxes containing the foci of said conics lie in said vertical principalplane and intersect normally said principal curve, the curvature of saidconic transverse sections at said principal curve below said great arcincreasing in a continuous and regular manner at a rate substantiallyequal to the accelerated rate of increase in curvature of the principalcurve itself, every transverse section of said variable front surfaceabove said great arc by planes orthogonal to said principal curve beingconics of eccentricity zero or greater, the lines of intersection ofsaid orthogonal planes and said vertical principal planes allintersecting at the common center of curvature of said conics when theireccentricity is zero, the axes of said conics which contain their fociwhen their eccentricities are greater than zero coinciding with theirrespective lines of intersection of said orthogonal planes and saidvertical principal plane, the eccentricities of said conic transversesection below said great arc increasing continuously and regularly withdistance from said great arc, all transverse sections by planesorthogonal to the principal curve being conics of eccentricity greaterthan zero when said great arc is a conic of eccentricity greater thanzero, the derivative of curvature of all of said conic transversesections of eccentricities greater than zero vanishing at said principalcurve, there being sufficient thickness of said lens to allow for thegenerating, grinding and polishing of a coacting coicoid back surface ofeccentricity zero or greater or toric back surface to incorporate anophthalmic prescription into said lens.
 2. A lens as defined in claim 1,wherein said great arc is circular and the portion of the principalcurve above said circular great arc is also circular and of the sameradius of curvature as the circular great arc, all of said conictransverse sections by planes orthogonal to said circular portion of theprincipal curve being circular and of the same radius of curvature asthe circular great arc, the derivative of curvature of the variablefront surface vanishing in all meridian sections about all normals tosaid variable surface along said circular great arc, the derivative ofcurvature of all of said conic transverse sections of eccentricitesgreater than zero by planes orthogonal to the remaining portion of theprincipal curve vanishing at said principal curve.
 3. A lens as definedin claim 1, wherein said principal curve at least below said great arcis elliptical, with its oblate point at said axial umbilical point andthe minor axis of said elliptical portion of said principal curvecoinciding with the axis of said variable front surface, said great arcis circular, and said lens can be osculated tangentially by a sphericalsurface of the same length radius of curvature as that of said circulargreat arc all along said circular great arc without said osculatingspherical surface crossing or intersecting said variable front surface.4. A lens as defined in claim 1, wherein said principal curve at leastbelow said great arc is elliptical, with its oblate point at said axialumbilical point and the minor axis of said elliptical portion of saidprincipal curve coinciding with the axis of said variable front surface,said great arc is elliptical, with its prolate point at said axialumbilical point and with the major axis of said elliptical great arccoinciding with the axis of said variable front surface, all transversesections by planes orthogonal to the principal curve being conics ofeccentricity greater than zero, and said lens can be osculatedtangentially by a prolate ellipsoid of revolution whose major axiscoincides with the axis of said variable surface and whose apical radiusof curvature and eccentricity are the same as that of said ellipticalgreat arc, all along said elliptical great arc, without said osculatingellipsoid of revolution crossing or intersecting said variable frontsurface.
 5. A lens as defined in claim 1, wherein said great arc iselliptical with the major axis of said elliptical great arc coincidingwith the axis of said variable surface and with the portion of theprincipal curve above said elliptical great arc also elliptical andhaving the same apical radius of curvature and eccentricity as saidelliptical great arc, the major axis of said portion of the principalcurve above said elliptical great arc coinciding with the axis of thevariable surface, all of said conic transverse sections by planesorthogonal to said elliptical upper portion of the principal curve beingelliptical arcs whose major areas coincide with the lines ofintersection of said orthogonal planes and said vertical principalplane, the curvature of said conic transverse sections along said upperportion of the principal curve decreasing at an accelerated rate withincreasng distance from said elliptical great arc, said accelerated rateof decrease in curvature being less than that of said upper portion ofthe principal curve itself, the eccentricities of said elliptical arctransverse sections also decreasing with increasing distance from saidelliptical great arc, the derivative of curvature of all sectionsorthogonal to said elliptical great arc vanishing at said great arc, andthe derivative of curvature of all conic transverse sections by planesorthogonal to the principal curve vanishing at said principal curve. 6.A lens as defined in claim 1, wherein the great arc is circular and saidprincipal curve is elliptical with its minor axis coinciding with theaxis of said variable surface, all of said transverse sections above andbelow said circular great arc by planes orthogonal to said principalcurve being conics of eccentricites greater than zero whose axescontaining their foci lie in said vertical principal plane and intersectnormally said principal curve, with said variable surface symmetricalabout said great arc, the derivative of curvature of the variablesurface vanishing or zero in all meridian sections about all normals tosaid variable surface along said circular great arc, and the derivativeof curvature of all of said conic transverse sections by planesorthogonal to said principal curve, above and below said circular greatarc, vanishing at said principal curve.
 7. A lens as defined in claim 1,wherein said great arc is circular and said principal curve is formed ofone elliptical arc below said great arc and another elliptical arc abovesaid great arc, the minor axes of both said elliptical arcs coincidingwith the axis of said variable surface, all of said transverse sectionsof said variable surface, above said circular great arc, by planesorthogonal to said principal curve being conics of eccentricity greaterthan zero, whose axes containing their foci lie in said verticalprincipal plane and intersect normally said principal curve, thecurvature of said conic transverse sections along said principal curveabove said circular great arc increasing at an accelerated rate withincreasing distance from said circular great arc, said accelerated rateof increase in curvature being substantially equal to the acceleratedrate of increase in curvature along the principal curve itself, theeccentricites of said conic transverse sections above said circulargreat arc also increasing with distance from said circular great arc,the derivative of curvature of the variable surface vanishing or zero inall meridian sections about all normals to said variable surface alongsaid circular great arc, and the derivative of curvature of all of saidconic transverse sections by planes orthogonal to said principal curve,above and below said circular great arc, vanishing at said principalcurve.
 8. A lens as defined in claim 1, wherein said great arc iselliptical with the major axis of said elliptical great arc containingwith the axis of said variable surface and said principal curve iselliptical with its major axis coinciding with the axis of said variablesurface, all of said transverse sections above and below said ellipticalgreat arc by planes orthogonal to said principal curve being conics ofeccentricities greater than that of said elliptical great arc and whosemajor axis containing the foci of said conics lie in said verticalprincipal plane and intersect normally said principal curve, with saidvariable surface symmetrical about said great arc, the derivative ofcurvature of all sections by planes orthogonal to said elliptical greatarc vanishing at said elliptical great arc, and the derivative ofcurvature of all conic transverse sections by planes orthogonal to saidprincipal curve vanishing at said principal curve.
 9. A lens as definedin claim 1, wherein said great arc is elliptical with the major axis ofsaid great arc coinciding with the axis of the variable surface and saidprincipal curve is formed of one elliptical arc below said ellipticalgreat arc and another elliptical arc above said elliptical great arc,the minor axes of both said elliptical arcs comprising said principalcurve coinciding with the axis of said variable surface, all of saidtransverse sections of said variable surface above and below saidelliptical great arc by planes orthogonal to said principal curve beingconics of eccentricities greater than that of said elliptical great arcand whose major axes containing the foci of said conics lie in saidvertical principal plane and intersect normally said principal curve,the curvature of said conic transverse sections along said principalcurve above said elliptical great arc increasing at an accelerated ratewith increasing distance from said elliptical great arc, saidaccelerated rate of increase in curvature being substantially equal tothe accelerated rate of increase in curvature along the principal curveitself, the eccentricities of said conic transverse sections above saidgreat arc increasing the distance from said great arc, the derivative ofcurvature of all sections by planes orthogonal to said elliptical greatarc vanishing at said great arc, and the derivative of curvature of allof said conic transverse sections by planes orthogonal to the principalcurve vanishing at said principal curve.
 10. A lens as defined in claim1 of dimensions sufficient for an ophthalmic spectacle lens.
 11. A lensas defined in claim 1 of a size approximately equal to or slightlysmaller than the human cornea for a corneal contact lens.